Blending modi gaussian closure and non-gaussian reduced subspace methods for turbulent dynamical systems, in preperation

Turbulent dynamical systems are characterized by persistent instabilities which are bal-anced by nonlinear dynamics that continuously transfer energy to the stable modes. To model this complex statistical equilibrium in the context of uncertainty quanti\u85cation all dynamical components (unstable modes, nonlinear energy transfers, and stable modes) are equally crucial. Thus, order-reduction methods present important limitations. On the other hand UQ methods based on the tuning of the non-linear energy uxes using steady state information (such as the modi ed quasilinear Gaussian (MQG) closure) may present discrepancies in extreme excitation scenarios. In this paper we derive a blended framework that links inexpensive second-order UQ schemes that model the full space (such as MQG) with high order statistical models in speci\u85c reduced-order subspaces. The coupling occurs in the energy transfer level by i) correcting the nonlinear energy uxes in the full space using reduced subspace statistics, and ii) by modifying the reduced-order equations in the subspace using information from the full space model. The results are illustrated in two strongly unstable systems under extreme excitations. The blended method allows for the correct prediction of the second-order statistics in the full space and also the correct modeling of the higher-order statistics in reduced-order subspaces.